User blog:Rgetar/Definitions update
Due to changes (1, 2, 3) of my array notation (1, 2, 3) it is needed to update definitions. Arrays are written as sequence of ⟨c⟩e pairs, separated with commas (,), where c is coordinates of element, and e is element at this coordinates. Coordinates also can be arrays. Sometimes coordinates can be omitted. If an array is not zero, all its zero elements may be omitted. Arrays are written in order of decreasing of coordinates. And, if coordinates are arrays, first compared elements with larger coordinates, then with smaller. Example of array: 1, 6, 0, 0, 3, 0, 0, 0, 5, 0 = ⟨9⟩1, ⟨8⟩6, ⟨7⟩0, ⟨6⟩0, ⟨5⟩3, ⟨4⟩0, ⟨3⟩0, ⟨2⟩0, ⟨1⟩5, ⟨0⟩0 = ⟨9⟩1, ⟨8⟩6, ⟨5⟩3, ⟨1⟩5 = ⟨9⟩1, 6, ⟨5⟩3, ⟨1⟩5 Functions of array 1. eo means "element of" eo(X; Y) is element of array X at coordinates Y 2. est means "element set to" est(X; Y; α) is array X with element at coordinates Y set to α 3. beo means "base element of" If array ≠ 0 then its base elements are non-zero elements. If array = 0 then its base element is element at coordinates 0, that is 0. Base elements are enumerated from right to left beginning from 1. beo(X; n) is n-th base element of array X 4. best means "base element set to" best(X; n; α) is array X with n-th base element set to α 5. cobeo means "coordinates of base element of" cobeo(X; n) is coordinates of n-th base element of array X 6. cobest means "coordinates of base element set to" cobest(X; n; Y) is array X with coordinates of n-th base element set to Y 7. nobe means "number of base elements" nobe(X) is number of base elements of array X 8. sobe means "set of base elements" sobe(X) is set of base elements of array X (old designation: sonze(X; Y)) 9. isobe means "iterated set of base elements" isobe(X) is sobe(X) ∪ {isobe(cobeo(X; n))}, 1 ≤ n ≤ nobe(X) (old designation: isonze(X; Y)) 10. leo means "last element of" leo(X) is eo(X; 0) 11. lest means "last element set to" lest(X; α) is est(X; 0; α) 12. clest means "conditionally last element set to" \(clest(X; α) = \left\{\begin{array}{lcr} X \qquad \qquad \, \text{if} \; α = 0\\ lest(X; α) \quad \text{if} \; α ≠ 0\\ \end{array}\right. \) 13. X0 X0 = lest(X; 0) 14. X-1 If leo(X) = α + 1 then X-1 = lest(X; α) 15. X--1 If lbeo(X) = α + 1 then X--1 = lbest(X; α) 16. fbeo means "first base element of" fbeo(X) is beo(X; nobe(X)) 17. lbeo means "last base element of" lbeo(X) is beo(X; 1) 18. cofbeo means "coordinates of first base element of" cofbeo(X) is cobeo(X; nobe(X)) (old designation: cofrewnzeloi(X)) 19. colbeo means "coordinates of last base element of" colbeo(X) = X' is cobeo(X; 1) Note: X" = (X')' = colbeo2(X) X"' = ((X')')' = colbeo3(X) etc. 20. lrt means "left rest" lrt(X) is array X without its last base element X = lrt(X), ⟨X'⟩lbeo(X) 21. rrt means "right rest" rrt(X) is array X without its first base element X = ⟨cofbeo(X)⟩fbeo(X), rrt(X) 22. ileo means "iterated last element of" ileo(X) is leo(X) ∪ ileo(X') 23. ilbeo means "iterated last base element of" ilbeo(X) is lbeo(X) ∪ ilbeo(X') 24. fbest means "first base element set to" fbest(X; α) is best(X; nobe(X); α) 25. lbest means "last base element set to" fbest(X; α) is best(X; 1; α) 26. cofbest means "coordinates of first base element set to" cofbest(X; Y) is cobest(X; nobe(X); Y) 27. colbest means "coordinates of last base element set to" colbest(X; Y) is cobest(X; 1; Y) 28. Note: X* is now invalid and is not used. Additional designations (X; a; b) \((X; a; b) = \left\{\begin{array}{lcr} a \quad \text{if} \; X' ≠ 0\\ b \quad \text{if} \; X' = 0\\ \end{array}\right. \) or \((X; a; b) = \left\{\begin{array}{lcr} a \quad \text{if} \; X\{·\}a \; \text{depends on} \; a\\ b \quad \text{if} \; X\{·\}a \; \text{does not depend on} \; a\\ \end{array}\right. \) Negative coordinates Elements of array with negative coordinates (⟨-α⟩) should be ignored. Identity function -1a = a X{·}a Short form X{·}a = {lbest(X; β), (X'; 1; a)}, β < lbeo(X), Y ∈ X'{·}a Expanded form (Without additional designations (X; a; b) and ⟨-α⟩): \(X\{·\}a = \left\{\begin{array}{lcr} \{lbest(X; β)\} \quad \text{if} \; X' = 0\\ \left.\begin{array}{lcr} \{lbest(X; β), \langle Y \rangle a\} \quad \text{if} \; X'^\prime = 0\\ \{lbest(X; β), \langle Y \rangle 1\} \quad \text{if} \; X'^\prime ≠ 0\\ \end{array}\right\} \; \text{if} \; X' ≠ 0\\ \end{array}\right. \quad ,β < lbeo(X), \; Y ∈ X'\{·\}a \) Ordinal array function Xa Short form 0a = a + 1 Xa = sup(-1; X0)Ya), Y ∈ X{·}a Expanded form (Without additional designations (X; a; b) and -1a): \(0a = a + 1\) \(Xa = \left\{\begin{array}{lcr} sup(X^0Ya) \quad \ \text{if} \; X' = 0\\ sup(Ya) \qquad \quad \text{if} \; X' ≠ 0\\ \end{array}\right. \quad ,X ≠ 0, \; Y ∈ X\{·\}a \) Generalized Veblen function φ(X) = α is (1 + leo(X))-th common fixed point of all functions α = φ(Y), Y ∈ X0{·}α Fundamental sequences Here is fundamental sequence system (6th) for generalized Veblen function. δ \(δ = \left\{\begin{array}{lcr} 0 \qquad \qquad \qquad \, \text{if} \; leo(X) = 0\\ φ(X-1) + 1 \quad \text{if} \; leo(X) ≠ 0\\ \end{array}\right. \) Xnα (Here and below "l." means "limit ordinal", and "s." means "successor ordinal"). \(Xn_α = \left\{\begin{array}{lcr} lbest(X; lbeo(X)n) \quad \text{if} \; lbeo(X) \; \text{- l.}\\ \left.\begin{array}{lcr} X--1, \langle X'n_α\rangle 1 \quad \text{if} \; leo(X') \; \text{- not s.}\\ X--1, \langle X'-1\rangle α \quad \text{if} \; leo(X') \; \text{- s.}\\ \end{array}\right\} \; \text{if} \; lbeo(X) \; \text{- s.}\\ \end{array}\right. \) φ(X)n To get fundamental sequence of Cantor normal form, replace its last term with fundamental sequence of the last term. Fundamental sequence for Cantor normal form term: \(φ(X)n = \left\{\begin{array}{lcr} φ(X-1)·n \quad \text{if} \; X \; \text{- s.}\\ \left.\begin{array}{lcr} φ(Xn_0) \quad \text{if} \; leo(X) \; \text{- l.}\\ \left.\begin{array}{lcr} φ(lest(X^0n_0; δ)) \quad \text{if} \; ilbeo(X_0) ∋ \; \text{l.} \; or \; n=0\\ φ(X^0n_{φ(X)n-1}) \quad \text{if} \; ilbeo(X_0) ∌ \; \text{l.} \; and \; n>0\\ \end{array}\right\} \; \text{if} \; leo(X) \; \text{- not l.}\\ \end{array}\right\} \; \text{if} \; X \; \text{- not s.}\\ \end{array}\right. \) Or, in a more understandable form: \(φ(X)n = \left\{\begin{array}{lcr} φ(X-1)·n \quad \text{if} \; X \; \text{- s.}\\ \left.\begin{array}{lcr} φ(lest(X; leo(X)n)) \quad \text{if} \; leo(X) \; \text{- l.}\\ \left.\begin{array}{lcr} φ(lest(X^0n_0; δ)) \quad \text{if} \; ilbeo(X_0) ∋ \; \text{l.} \; or \; n=0\\ φ(X^0n_{φ(X)n-1}) \quad \text{if} \; ilbeo(X_0) ∌ \; \text{l.} \; and \; n>0\\ \end{array}\right\} \; \text{if} \; leo(X) \; \text{- not l.}\\ \end{array}\right\} \; \text{if} \; X \; \text{- not s.}\\ \end{array}\right. \) Clested version: \(φ(X)n = \left\{\begin{array}{lcr} φ(X-1)·n \quad \text{if} \; X \; \text{- s.}\\ \left.\begin{array}{lcr} φ(Xn_0) \quad \text{if} \; leo(X) \; \text{- l.}\\ \left.\begin{array}{lcr} φ(clest(X^0n_0; δ)) \quad \text{if} \; ilbeo(X_0) ∋ \; \text{l.} \; or \; n=0\\ φ(X^0n_{φ(X)n-1}) \quad \text{if} \; ilbeo(X_0) ∌ \; \text{l.} \; and \; n>0\\ \end{array}\right\} \; \text{if} \; leo(X) \; \text{- not l.}\\ \end{array}\right\} \; \text{if} \; X \; \text{- not s.}\\ \end{array}\right. \) Older version (2nd): \(φ(X)n = \left\{\begin{array}{lcr} φ(X-1)·n \quad \text{if} \; X \; \text{- s.}\\ \left.\begin{array}{lcr} φ(Xn_0) \quad \text{if} \; leo(X) \; \text{- l.}\\ \left.\begin{array}{lcr} φ(lest(X^0n_0; δ)) \quad \text{if} \; ilbeo(X_0) ∋ \; \text{l.}\\ \left.\begin{array}{lcr} φ(X^0n_{φ(X)n-1}) \quad \text{if} \; n>0\\ φ(X^0n_δ) \qquad \qquad \text{if} \; n=0\\ \end{array}\right\} \; \text{if} \; ilbeo(X_0) ∌ \; \text{l.}\\ \end{array}\right\} \; \text{if} \; leo(X) \; \text{- not l.}\\ \end{array}\right\} \; \text{if} \; X \; \text{- not s.}\\ \end{array}\right. \) Clested version: \(φ(X)n = \left\{\begin{array}{lcr} φ(X-1)·n \quad \text{if} \; X \; \text{- s.}\\ \left.\begin{array}{lcr} φ(Xn_0) \quad \text{if} \; leo(X) \; \text{- l.}\\ \left.\begin{array}{lcr} φ(clest(X^0n_0; δ)) \quad \text{if} \; ilbeo(X_0) ∋ \; \text{l.}\\ \left.\begin{array}{lcr} φ(X^0n_{φ(X)n-1}) \quad \text{if} \; n>0\\ φ(X^0n_δ) \qquad \qquad \text{if} \; n=0\\ \end{array}\right\} \; \text{if} \; ilbeo(X_0) ∌ \; \text{l.}\\ \end{array}\right\} \; \text{if} \; leo(X) \; \text{- not l.}\\ \end{array}\right\} \; \text{if} \; X \; \text{- not s.}\\ \end{array}\right. \) See also other (1st, 3rd, 4th, 5th, modified and clested) versions. 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